Answer
$$-12 $$
Work Step by Step
Given $$\lim _{\theta \rightarrow 0} \frac{\cos 7 \theta-\cos 5 \theta}{\theta^{2}}$$ Consider $$ f(\theta)= \frac{\cos 7 \theta-\cos 5 \theta}{\theta^{2}}$$ From the following figure, we can observe that \begin{align*} \lim _{\theta\rightarrow 0^+} f( \theta)&= -12\\ \lim _{\theta \rightarrow 0^-} f( \theta)&= -12 \end{align*} Then $$\lim _{\theta \rightarrow 0} \frac{\cos 7 \theta-\cos 5 \theta}{\theta^{2}}=-12 $$
